Assignment

Math Probability Problem (Optional Challenge)

There is an urn containing three balls. Each ball is either red or blue, and two balls have the same color. So, there are two possibities: either two balls are red and one is blue (the “majority-red case”) or two balls are blue and one is red (the “majority-blue” case). The probability of both cases is the same, i.e., 50%.
John is allowed to draw a random ball from the urn, look at it, and put it back. Then, he has to make a guess as to whether he thinks the urn is majority-blue or majority-red. Let’s say that John has drawn a red ball. It seems
natural that John should guess that the urn is probably majority-red. We now ask you to confirm this is a good
guess, using Bayes’ rule.

Let “MajRed” be the event that the urn is majority-red. So, “not MajRed” is the event that the urn is not majorityred, which means that the urn is majority-blue. Let “RedBall” be the event that John picked a red ball. We are interested in the probability P(MajRed|RedBall) that the urn is majority-red given that John has picked a red ball.

(a) What is the prior probability P(MajRed) according to the above description?

(d) Finally, using Bayes’ rule and other rules, what is the conditional probability P(MajRed | RedBall)?

John decided to declare that he thinks the urn is majority-red. Next, it’s Mary’s turn to pick a random ball from the rn, look at it, and put it back. She picks a blue ball. Now, she is in the same situation as John: she has to make a guess about the urn based on what she has seen. But, she also knows that John has seen a red ball (because otherwise he would not have said that he thinks the urn is majority red).

Mary is not sure what to think. On the one hand, she saw a blue ball, and therefore the urn is most likely majorityblue. On the other hand, she knows that John has picked a red ball, which suggests that the urn is majority-red.

We can solve this problem again using Bayes’ rule. Let “RedThenBlue” be the event that John picked a red ball,
and then Mary picked a blue ball (which is what happened).